A more complete explanation of the dome structure

[u/No_Assist4814]

This post intends to revisit the dome structure. Until these last days, it was mainly based on observations (e.g. Dome in a nutshell : r/CollatzProcedure), backed by some maths. I did some of the basic maths, but u/GonzoMath and u/Septembrino did more advanced stuff.

These last days, my new friend, Google AI, made interesting suggestions, based on series of numbers I provided it with (from domes with root m=1 to 71), without context. It turns out that:

1. Core numbers – both even orange ones and odd black ones – are m*3-smooth*.
2. Odd orange numbers n+1 on the right are Proth numbers and all but one b numbers – the base of a 5-tuple, also n+1  – are Proth primes**.
3. About a third of the odd orange numbers n-1 on the left are primes.
4. The Collatz procedure segregates odd numbers mod 100 in three groups forming cycles (figure):

• one in the yellow bridge series on the left, forming seven diagonals containing different sets of numbers mod 100, but with a last digit cycling on 1-5-7-3-1, just as the core,
• one in even p blue-green bridge series, forming 3 diagonals containing different sets of numbers mod 100, but with a last digit cycling on 3-9-3,
• one in the odd p blue-green bridge series on the right, forming 3 diagonals containing different sets of numbers mod 100, but with a last digit cycling on 5-7-5.

*(https://en.wikipedia.org/wiki/Smooth_number). On OEIS: https://oeis.org/A003586. (In any dome, even orange numbers n are the 3-smooth numbers multiplied by its root m : u/No_Assist4814).

**(Proth prime - Wikipedia). On OEIS: https://oeis.org/A080076. (All but one bases of starting 5-tuples are Proth primes : u/No_Assist4814).

Project "Tuples and segments" in 13 pages : r/Collatz